### 5.1 Near-extremal Kerr–Newman black holes

Near-extremal Kerr–Newman black holes are characterized by their mass , angular momentum
and electric charge . (We take without loss of generality.) They contain
near-extremal Kerr and Reissner–Nordström black holes as particular instances. The metric
and thermodynamic quantities can be found in many references and will not be reproduced
here.
The near-extremality condition (145) is equivalent to the condition that the reduced Hawking
temperature is small,

Indeed, one has and the term in between the brackets is of
order one since , and . Therefore, we can use
interchangeably the conditions (145) and (147).
Since there is both angular momentum and electric charge, extremality can be reached both in the
regime of vanishing angular momentum and vanishing electric charge . When angular momentum is
present, we expect that the dynamics could be described by the , while when electric charge is
present the dynamics could be described by the . It is interesting to remark that the condition

implies (145) – (147) since but it also implies . Similarly, the condition
implies (145) – (147), since , but it also implies . In the following, we will need
only the near-extremality condition (145), and not the more stringent conditions (148) or (149). This is the
first clue that the near-extremal scattering will be describable by both the and the
.
Near-extremal black holes are characterized by an approximative near-horizon geometry, which controls
the behavior of probe fields in the window (146). Upon taking and taking the limit
the near-horizon geometry decouples, as we saw in Section 2.7.

Probes will penetrate the near-horizon region close to the superradiant bound (146). When
we need

Indeed, repeating the reasoning of Section 2.6, we find that the Boltzman factor defined in the near-horizon
vacuum (defined using the horizon generator) takes the following form
where , and are the quantum numbers defined in the exterior asymptotic region and
is finite upon choosing (150). The conclusion of this section is that the geometries (79) control the
behavior of probes in the near-extremal regime (145) – (146). We identified the quantity as a
natural coefficient defined near extremality. It will have a role to play in later Sections 5.3 and
5.4. We will now turn our attention to how to solve the equations of motion of probes close to
extremality.